How to Find the Vertex of a Parabola on the SAT (Formula & Steps)

The SAT tests vertex questions two ways: for standard form y = ax² + bx + c, use the vertex formula x = −b/(2a) and plug that x back in for y; for vertex form y = a(x − h)² + k, the vertex is simply (h, k). Both types appear in the Advanced Math domain and both usually take under a minute with Bluebook's built-in Desmos calculator.

Below are two SAT-style worked examples, the method, the two question phrasings you will actually see, and the Desmos shortcut that saves time on the day.

Worked example 1: y = 2x² − 8x + 5 (standard form)

SAT-style problem: The function f is defined by f(x) = 2x² − 8x + 5. What is the y-coordinate of the vertex of the graph of y = f(x) in the xy-plane?

1. Identify a, b, c: a = 2, b = −8, c = 5.
2. Vertex x-coordinate: x = −b/(2a) = −(−8)/(2·2) = 8/4 = 2.
3. Plug x = 2 back into f: f(2) = 2(2)² − 8(2) + 5 = 8 − 16 + 5 = −3.
4. Vertex is (2, −3), so the y-coordinate is −3.

Because a > 0, this parabola opens upward and (2, −3) is a minimum. If the SAT asked for the minimum value of f instead, the answer is the same −3.

The vertex method on the SAT, step by step

1. Recognize the form. Standard: y = ax² + bx + c. Vertex: y = a(x − h)² + k. Factored: y = a(x − p)(x − q) — vertex x is the midpoint (p + q)/2.
2. Standard form → use x = −b/(2a). Compute x, then plug back in to get y. The pair is the vertex.
3. Vertex form → read (h, k) directly. Do not solve — the vertex is baked into the equation. Watch the sign (next section).
4. Answer what the SAT actually asked. The stem may want the x-coordinate, the y-coordinate, the ordered pair, or the minimum/maximum value. Minimum/maximum is just the y.

The sign trap the SAT loves in vertex form

Vertex form is y = a(x − h)² + k, so the sign inside the parenthesis is a subtraction. That means y = 3(x + 5)² − 7 puts h = −5, not +5, and the vertex sits at (−5, −7). Roughly half of missed SAT vertex questions are this exact sign flip. If it helps, rewrite the equation as 3(x − (−5))² − 7 before reading off h.

Worked example 2: y = −x² + 6x − 4 (find the max)

SAT-style problem: The function g is defined by g(x) = −x² + 6x − 4. What is the maximum value of g?

1. Identify a, b, c: a = −1, b = 6, c = −4. Because a < 0, the parabola opens downward and the vertex is a maximum.
2. Vertex x-coordinate: x = −b/(2a) = −6/(2·(−1)) = −6/(−2) = 3.
3. Maximum value: g(3) = −(3)² + 6(3) − 4 = −9 + 18 − 4 = 5.
4. The maximum value of g is 5.

Cross-check by completing the square: g(x) = −(x² − 6x) − 4 = −((x − 3)² − 9) − 4 = −(x − 3)² + 5. Vertex form confirms the max value is 5 at x = 3. ✓

"Which equation shows the vertex as constants?" — the SAT's other vertex question

A common SAT vertex question gives you a parabola in standard form and four equivalent equations, then asks which one displays the coordinates of the vertex as constants or coefficients. Every choice is the same parabola written differently; the correct one is vertex form. The move is completing the square.

For y = 2x² − 8x + 5: factor the leading coefficient out of the x-terms (y = 2(x² − 4x) + 5), complete the square inside (y = 2((x − 2)² − 4) + 5), and simplify (y = 2(x − 2)² − 3). Now the vertex (2, −3) is visible as the constants (h, k).

Bluebook Desmos shortcut

Bluebook ships with Desmos. On any vertex question with clean numeric coefficients, paste the equation into Desmos, click the turning point, and read the coordinates off the graph — about 20 seconds. It stops helping on abstract questions (if the vertex of y = ax² + bx + c is at (h, k)…) where the SAT wants a symbolic answer, not a graphed one.

Common mistakes on SAT vertex questions

• Reading (x + 5) as h = +5. Vertex form has a built-in minus sign, so (x + 5)² means h = −5. Sign flip = wrong answer.
• Answering the x when the SAT asked for y. Always re-read the stem after finding the vertex. "y-coordinate" and "minimum value" both want y.
• Forgetting to plug x back in. The vertex is a point. If the question wants the min or max value, stopping at x is zero credit.
• Dropping the leading coefficient in completing the square. If a ≠ 1, factor a out of the x-terms first. Otherwise your k is wrong by a factor of a.
• Trusting Desmos on abstract questions. Desmos graphs numeric equations. Anything with parameters needs algebra.
• Mixing max and min. If a > 0 the vertex is a minimum; if a < 0 it is a maximum. Getting this backward on a "which of the following describes the graph" question is an easy miss.

For a general-math walkthrough of the vertex formula without SAT framing, see our companion post: How to Find the Vertex of a Parabola (Formula & Examples). If quadratics are the domain you keep missing, how to find the discriminant of a quadratic equation covers the other Advanced Math workhorse — b² − 4ac tells you how many real solutions a quadratic has, another common SAT stem.

The formula and the method above will get you every vertex question the SAT throws at you. To see how many you would get right today under real conditions — and which other Advanced Math topics are dragging your score — take the free SAT math diagnostic above. Up to 15 questions, no account required to start.